Stability and convergence of a time-fractional variable order hantush equation for a deformable aquifer

Abstract and Applied Analysis

Volumn 2013, Issue , 2013, Pages

  Atangana, Abdon ^a   Oukouomi Noutchie, S C ^b  

^a UNIVERSITY OF THE FREE STATE   (South Africa)

^b NORTH WEST UNIVERSITY   (South Africa)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84879318476     PISSN: 10853375     EISSN: 16870409     Source Type: Journal    
DOI: 10.1155/2013/691060     Document Type: Article

References (23)

1. Kruseman G. P., de Ridder N. A., Analysis and Evaluation of Pumping Test Data 1990 2nd Wageningen, The Netherlands International Institute for Land Reclamation and Improvement
2. Theis C. V., The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Transactions, American Geophysical Union 1935 16 2 519 524
3. Hantush M. S., Analysis of data from pumping tests in leaky aquifers. Transactions, American Geophysical Union 1956 37 6 702 714
4. Hantush M. S., Jacob C. E., Non-steady radial flow in an infinite leaky aquifer. Transactions, American Geophysical Union 1955 36 1 95 100 MR0072361
5. Umarov S., Steinberg S., Variable order differential equations with piecewise constant order-function and diffusion with changing modes. Journal of Analysis and Its Applications 2009 28 4 431 450 10.4171/ZAA/1392 MR2550698 ZBL1181.35359
6. Ross B., Samko S., Fractional integration operator of variable order in the Hölder spaces H (x). International Journal of Mathematics and Mathematical Sciences 1995 18 4 777 788 10.1155/S0161171295001001 MR1347069 ZBL0838.26005
7. Atangana A., Botha J. F., Generalized groundwater flow equation using the concept of variable order derivative. Boundary Value Problems 2013 2013, article 53 10.1186/1687-2770-2013-53
8. Atangana A., Klçman A., A possible generalization of acoustic wave equation using the concept of perturbed derivative order. Mathematical Problems in Engineering 2013 2013 6 696597 10.1155/2013/696597
9. Zhang Y., A finite difference method for fractional partial differential equation. Applied Mathematics and Computation 2009 215 2 524 529 10.1016/j.amc.2009.05.018 MR2561509 ZBL1177.65198
10. Tadjeran C., Meerschaert M. M., Scheffler H.-P., A second-order accurate numerical approximation for the fractional diffusion equation. Journal of Computational Physics 2006 213 1 205 213 10.1016/j.jcp.2005.08.008 MR2203439 ZBL1089.65089
11. Meerschaert M. M., Tadjeran C., Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathematics 2004 172 1 65 77 10.1016/j.cam.2004.01.033 MR2091131 ZBL1126.76346
12. Deng W. H., Numerical algorithm for the time fractional Fokker-Planck equation. Journal of Computational Physics 2007 227 2 1510 1522 10.1016/j.jcp.2007.09.015 MR2442403
13. Li C. P., Chen A., Ye J. J., Numerical approaches to fractional calculus and fractional ordinary differential equation. Journal of Computational Physics 2011 230 9 3352 3368 10.1016/j.jcp.2011.01.030 MR2780467 ZBL1218.65070
14. Chen C.-M., Liu F., Turner I., Anh V., A Fourier method for the fractional diffusion equation describing sub-diffusion. Journal of Computational Physics 2007 227 2 886 897 10.1016/j.jcp.2007.05.012 MR2442379 ZBL1165.65053
15. Chen Y. Q., Moore K. L., Discretization schemes for fractional-order differentiators and integrators. IEEE Transactions on Circuits and Systems I 2002 49 3 363 367 10.1109/81.989172 MR1890973
16. Yuste S. B., Acedo L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM Journal on Numerical Analysis 2005 42 5 1862 1874 10.1137/030602666 MR2139227 ZBL1119.65379
17. Podlubny I., Chechkin A., Skovranek T., Chen Y. Q., Vinagre Jara B. M., Matrix approach to discrete fractional calculus. II. Partial fractional differential equations. Journal of Computational Physics 2009 228 8 3137 3153 10.1016/j.jcp.2009.01.014 MR2509311 ZBL1160.65308
18. Hanert E., On the numerical solution of space-time fractional diffusion models. Computers and Fluids 2011 46 33 39 10.1016/j.compfluid.2010.08.010 MR2948977
19. Zhang Y., A finite difference method for fractional partial differential equation. Applied Mathematics and Computation 2009 215 2 524 529 10.1016/j.amc.2009.05.018 MR2561509 ZBL1177.65198
20. Lin R., Liu F., Anh V., Turner I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Applied Mathematics and Computation 2009 212 2 435 445 10.1016/j.amc.2009.02.047 MR2531970 ZBL1171.65101
21. Sun H., Chen W., Li C., Chen Y., Finite difference schemes for variable-order time fractional diffusion equation. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2012 22 4 16 1250085 10.1142/S021812741250085X MR2926061 ZBL1258.65079
22. Crank J., Nicolson P., A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Mathematical Proceedings of the Cambridge Philosophical Society 1947 43 1 50 67 MR0019410 ZBL0029.05901
23. Li C. P., Tao C. X., On the fractional Adams method. Computers and Mathematics with Applications 2009 58 8 1573 1588 10.1016/j.camwa.2009.07.050 MR2562405 ZBL1189.65142